We use successor features (SF)[1]} to capture the agent-environment interaction model since SFs summarize the dynamics induced by the environment when following a behaviour policy. The first step of building a generally reusable agent-environment interaction model is to learn cross-domain transferable successor features in order to extract a general agent-environment interaction model. For simplicity, we consider a random uniform behaviour policy \(\pi _{0}\)  for all environment domains. Recall that, SF is defined as the expected cumulative features of \(\phi \)  by following \(\pi _{0}\)  starting at a specific state. The SF of \((x^{i}, a)\)  in environment domain \(Y_{i}\)  is defined as:
\(\begin{split}\psi ^{\pi _{0},i}(x^{{i}}, a) ={ \mathbb {E}}^{\pi _{0}}_{(x, a, x^{\prime }) \sim \mathcal {T}_{i}} [\sum _{t^{\prime }=t}^{\infty } \gamma ^{t^{\prime }-t} \phi _{x_{t+1}} |X_{t}=x^{{i}}, A_{t}=a].\end{split}\) 
